Prove the following statements.1. If a set of vectors contains exactly 2 vectors and those vectors are scalar multiples of each other, thenthe set is linearly dependent.2. If a set of vectors contains at least 2 vectors that are scalar multiples of each other, then the set islinearly dependent.3. If a set contains p vectors in R" and p> n, then the set is linearly dependent.4. If a set of vectors contains the zero vector, then the set is linearly dependent.5. If a set contains exactly one vector and it is nonzero, then the set is linearly independent.6. A set of vectors is linearly independent if and only if at least one vector in the set can be written as alinear combination of other vectors in the set.

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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Find a real number k such that the following three vectors in R4 are not linearly independent. (1, 0, –3, 2)", (4, 2, 4, 1)", (18, 10, k, 1)"
1. Determine if these vectors are linearly independent or linearly dependent. If they are linearly independent show this. If they are linearly dependent, write a nontrivial linear combination (at least some coefficients need to be nonzero) of them being equal to 0. {4.0-6)}
The set of vectors {(1,1,0,0),(0,-1,0,0),(0,0,1,1),(1,0,1,1)} is linearly dependent. What is a valid expression of the last vector as a linear combination of the others? a. (1,0,1,1)=(1,1,0,0)+(0,-1,0,0)-(0,0,1,1) b. (1,0,1,1)=(1,1,0,0)+(0,-1,0,0)+(0,0,1,1) c. (1,0,1,1)=-(1,1,0,0)+(0,-1,0,0)+(0,0,1,1) d. (1,0,1,1)=2(1,1,0,0)+2(0,-1,0,0)+2(0,0,1,1)
2. If a set of vectors contains at least 2 vectors that are scalar multiples of each other, then the set is linearly dependent.
Question 5. Suppose that f is a Möbius transformation which fixes the real and imag- inary axes, i.e. f({ z = C: Im(z) = 0 }) = { z = C: Im(z)=0} and ƒ({ z € C: Re(z) = 0 }) = { z = C: Re(z) = 0}. Prove that f maps circles centred at the origin to circles centred at the origin (not necessarily of the same radius), i.e. for all r₁ > 0, there exists r2 > 0 such that ƒ({ z ≤ C : |z| = r₁ }) = { z € C : |Z| = r₂}. A geometric argument is sufficient, as long as you clearly state any properties of Möbius transformations that you use. You may assume, should you wish, that a circle is centred on a line if and only if it intersects the line at right angles.
4. ) Determine whether the following three vectors are linearly independent or inearly dependent. v₁ = (3,3, 0, 2), v₂ = (0, 2, 3, 1), V3 (3, 0, -2, -1). Hint: You can turn them into column vectors. =
Let u1, U2, ..., un be a set of vectors, and suppose there are nontrivial solutions to x11x2u2+ +xnun = 0 Explain why, if x is a free variable, you can always express u; in terms of the vectors corresponding to the basic variables.
There are 6 linearly independent vectors in R5. O True False
Show that a set of vectors, which contains a set of linearly dependent vectors, is linearly dependent. What is the analogous statement about linearly independent vectors?
Prove that two vectors are linearly dependent (taken as a set of vectors) if and only if one vector is multiple of the other.

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1. If a set of vectors contains exactly 2 vectors and those vectors are scalar multiples of each other, then the set is linearly dependent.